Chapter 8.1, Problem 124E

(a)

To determine

To graph: The table data using graphing utility.

Explanation of Solution

Given: The revenue $R_n$ (in millions of dollars) of Netflix from 2008 through 2013 are shown in the table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Plot the point on graph of given table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Where, y -axis represents revenue $R_n$ (in millions of dollars) and x -axis represents number of years n.

Let $n=8$ corresponding to 2008.

  

(b)

To determine

To find: The linear and quadratic model of the given data.

Answer to Problem 124E

Linear equation: $R_n=625.96n-3841.53$

Coefficient of determination for linear, $r^2=0.9779$

Quadratic equation: $R_n=34.7875n^2-104.58n-107.675$

Coefficient of determination for quadratic, $r^2=0.9843$

Explanation of Solution

Given: The revenue $R_n$ (in millions of dollars) of Netflix from 2008 through 2013 are shown in the table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Plot the point on graph of given table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Where, y -axis represents revenue $R_n$ (in millions of dollars) and x -axis represents number of years n.

Let $n=8$ corresponding to 2008.

  

Using graph and graphing utility the linear regression equation.

  $R_n=625.96n-3841.53$

Coefficient of determination, $r^2=0.9779$

Quadratic model of the given graph.

  $R_n=34.7875n^2-104.58n-107.675$

Coefficient of determination, $r^2=0.9843$

(c)

To determine

To choose: The best fir model for the given table.

Answer to Problem 124E

Quadratic equation: $R_n=34.7875n^2-104.58n-107.675$

Explanation of Solution

Given: The revenue $R_n$ (in millions of dollars) of Netflix from 2008 through 2013 are shown in the table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Using regression of table get two model.

Linear equation: $R_n=625.96n-3841.53$

Coefficient of determination for linear, $r^2=0.9779$

Quadratic equation: $R_n=34.7875n^2-104.58n-107.675$

Coefficient of determination for quadratic, $r^2=0.9843$

The coefficient of determination is nearer to 1 will choose best model.

Therefore, quadratic model is a better fit for the data.

(d)

To determine

To find: The revenue in 2017.

Answer to Problem 124E

$8168.1 millions

Explanation of Solution

Given: The revenue $R_n$ (in millions of dollars) of Netflix from 2008 through 2013 are shown in the table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Quadratic equation: $R_n=34.7875n^2-104.58n-107.675$

Therefore, quadratic model is a better fit for the data.

Since 2008 showing $n=8$ . So, 2017 will be $n=17$

So, put $n=17$ into quadratic model to get revenue in 2017.

  $\begin{array}{l}{R}_n=34.7875{\left(17\right)}^2-104.58\left(17\right)-107.675\\ R_{17}=8168.05\end{array}$

Thus, Revenue of Netflix in 2017 will be $8168.1 million.

(e)

To determine

To find: The year when revenue will reach 15 billion dollars.

Answer to Problem 124E

2022

Explanation of Solution

Given: The revenue $R_n$ (in millions of dollars) of Netflix from 2008 through 2013 are shown in the table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Quadratic equation: $R_n=34.7875n^2-104.58n-107.675$

Put $R_n=15000$ into quadratic model.

  $\begin{array}{l}15000=34.7875n^2-104.58n-107.675\\ 34.7875n^2-104.58n-15107.675=0\\ n=22.4\\ n\approx 22\end{array}$

Since 2008 corresponding to $n=8$

So, $n=22$ corresponding to 2022.

Thus, In 2022 revenue will be 15 billion dollars.

(f)

To determine

To find: The total revenue from 2008 through 2013.

Answer to Problem 124E

  $R_{\text{total}}=\$16386.1$

Same

Explanation of Solution

Given: The revenue $R_n$ (in millions of dollars) of Netflix from 2008 through 2013 are shown in the table.

  $\begin{array}{ccccccc}\text{Year}& 2008& 2009& 2010& 2011& 2012& 2013\\ \text{Revenue,}{R}_n& 1364.7& 1670.3& 2162.6& 3204.6& 3609.3& 4374.6\end{array}$

Total revenue from 2008 through 2013.

  $R_n=1364.7+1670.3+2162.6+3204.6+4374.6$

  $R_{\text{total}}=\$16386.1$

Using equation total revenue area under the curve.

Quadratic equation: $R_n=34.7875n^2-104.58n-107.675$

Total revenue, $R_{\text{Total}}={\displaystyle \sum _{n=8}^{13}{R}_n}$

  $\begin{array}{c}{R}_{\text{Total}}=R_8+R_9+R_{10}+R_{11}+R_{12}+R_{13}\end{array}$

     $\begin{array}{c}=1282.08+1768.89+2325.27+2951.23+3646.75+4411.87\end{array}$

     $\begin{array}{c}=16386.1\end{array}$

Both are approximately same.